Tables de Mortalité
Instituto de Seguros de Portugal
Le 10 mars 2008
1
Calculation of mathematical provisions
• Carried out on the basis of recognised actuarial
methods
• The mortality table used in the calculation should
be chosen by the insurance undertaking taking into
account the nature of the liability and the risk class
of the product
• No mortality table is prescribed
2
Calculation of mathematical provisions
• Longevity risk is mainly important in annuities and
in term assurance
• With respect to term assurance companies are very
conservative in the choice of the mortality table used to
calculate premiums and mathematical provisions (very
high mortality rates compared to observed rates)
• In new life annuity contracts companies adequate the
choice of mortality tables to the effects of mortality gains
projected from recent experience
3
Calculation of mathematical provisions
• In old annuity contracts that were written on the basis of old
mortality tables, actuaries regularly analyse the sufficiency
of technical basis and reassess the mathematical provisions
according to more recent mortality tables
• The relative weight of life annuity mathematical provisions
represents about 2% of total mathematical provisions from
the life business
4
Market Information to the Supervisor
Information on the Annual Mortality Recorded and on the
Annual Exposed-to-Risk (broken down by age and sex) on the
following types of Mortality Risk:
•Death Risk
Term Assurances
•Survival Risk
Pure Endowments
Endowments and Whole Life
“Universal Life” types of policy
“Unit-linked” and “Index-linked” types of policy
5
Market Information to the Supervisor
Information on the Annual Mortality Recorded and on the
Annual Exposed-to-Risk (broken down by age and sex) on the
following types of Mortality Risk: (follow up)
•Annuitants Risk
Annuities
Pension Funds Annuitant Beneficiaries
Number of Pension Fund Members
6
Supervisory process
• Responsible actuary report
• ISP’s mortality studies
• Static and dynamic mortality tables
• Publication of papers and special studies
• ISP analysis of suitability of mortality tables used
7
Supervisory process
• Responsible actuary report
• The responsible actuary should:
• comment on the suitability of the mortality tables
used for the calculation of the mathematical provision
• produce a comparison between expected and actual
mortality rates
• Whenever significant deviations exist, he should measure
the impact of using mortality tables that are better
adjusted to the experience and the evolutionary
perspectives of the mortality rates
8
ISP Supervisory Process
• Feed-Back information from the Supervisor
• Under the Life Business Risk Assessment, ISP conducts
independent research and runs various statistical methods
(deterministic and stochastic) to ascertain the Trend and Volatility
of the multiple variables and risk sources that affect the Life
Business:
• Each year, ISP issues a Report on the Portuguese Insurance
and Pension Funds Market in which it publishes Special
Studies intended to feed-back information onto the Insurance
Undertakings and their Responsible Actuaries on the above
mentioned risk sources, their possible modelling techniques and
the corresponding parameters.
9
Mortality Projections for Life Annuities (example)
The force of mortality (mx) may be expressed as the first derivative of the rate of mortality (qx):
d (q x )
q
d
with
mx =
= lim h x = lim h x
h d x = l x  l xh
h

0
h

0
dx
h
h  lx
(l  l )
(l  l )
( d )
d (l x )
= lim x  h x =  lim x x  h = l x  lim h x = l x  m x
h0
h0
h0 h  l
dx
h
h
x
d (l x )
= m x  l x
Mortality Table
Mortality Table
dx
1.000.000
1,000
105
95
100
90
85
80
75
70
65
60
55
50
0,000
45
0,100
40
age (x)
m x = tg (q )
m x = tg (q )
m x = tg (q )
0,200
35
0
0,300
30
100.000
0,400
25
200.000
0,500
20
300.000
mx
To determine the value of tg (q )In graph
One should take into account the fact that
The cathets of the triangles should be taken
In their correct scale
0,600
15
400.000
0,700
5
500.000
qx
0,800
10
600.000
0,900
0
700.000
Prob. of 1 individual of age x)
Dying over 1 year = q(x)
800.000
0
5
10
15
20
25
30
35
40
45
50
55
60
65
70
75
80
85
90
95
100
105
l(x) = Nº individuals alive at age (x)
900.000
age (x)
10
If a mortality trend follows a Gompertz Law, then
m x  t = m x  e k t
hence
m 
m x t
= e k t , then k  t = ln xt 
mx
 mx 
and also
1 m 
k =  ln xt 
t  mx 
If mortality were static, then the complete expectation of Life would be
mx
o
ex =
k
e
k



1Z
 Z e
1
1
z
e
mx
k
1
whereg
 z
mx
k
 dz ,
or, in summary
m 
f x 
o
 k 
ex =
k
with
 m x n



k
 mx 

 dz = g  ln    
 k  n = 1 n  n!
= 0,5772157...
Is the Euler constant
11
Mortality Projections for Life Annuities (example)
Let us suppose now, that for every age the force of mortality tends to dim out as
time goes by, in such a way that an individual which t years before had age x
and was subject to a force of mortality mx , is now aged x+t and is subject
to a force of mortality lower than mx+t (from t years ago). The new force
 r t
of mortality will now be:
mt = m 
e
xt
r
e
xt
Where
translates the annual averaged relative decrease in the force of
mortality for every age
If we further admit another assumption, that the size relation between the forces
of mortality in successively higher ages is approximately constant over time,
k t
k t
i.e.:
t
t
and
m x t  m x 
m x t  m x 
then
m
t
x t
k t
 m e
t
x
e
 (m
hence
 r t
x
e
) e
k t
e
( k  r )t
= mx e
( k  r )t
m xt t  m x  e
12
John H. Pollard –“Improving Mortality: A Rule of Thumb and Regulatory Tool” – Journal of Actuarial Practice Vol. 10, 2002
Mortality Projections for Life Annuities (example)
The prior equation also implies that:
(k  r )t
e
m

mx
t
x t
k t
k t
m xt  e
m xt  e
k t
 r t
 r t
 e e 
e 
k t
mx
m x e
 m xt 
where  r  t  ln 
 m 
 x
hence, finally
 1  m xt 
r
 ln  
t
 mx 
13
Mortality
Projections for Life Annuities (example)
The practical application of the theoretical concepts involving the variables k
and r may be illustrated in the graph bellow:
q x [T  t ]
q x [T ]
1,000
0,900
m x [T  t ]
m x [T ]
0,800
0,700
m x t = m x  e k  t
0,600
0,500
m tx = m x  e  rt
0,400
0,300
0,200
0,100
age (x)
110
105
100
95
90
85
80
75
70
65
60
55
50
45
40
35
30
25
20
15
10
5
0,000
0
Prob. of 1 individual aged (x)
dying in the course of 1 year = q(x)
Mortality Tables
14
Mortality Projections for Life Annuities (example)
In order to increase the “goodness of fit” of the mortality data by using the theoretical
Gompertz Law model involving the variables k and r, it is sometimes best to
assume that r has different values for different age ranges (we may, for example,
use r1 for the younger ages and r2 for the older ages)
Mortality of insured lives of the survival-risk-type of life assurance in Portugal:
2000-2002 (Males )
qx
0,1500
upper boundary
0,1300
( Spline 1997 )
2001 )
m x(Gompertz
= m 36
e


1
2
0,1100
0,0900
0,0700
(Gompertz
m x
1
2
2001 )
( Spline 1997 )
= m 20  1
2
k (51100 years ) ( x  36 )
1
2
k ( 20 50 years )( x  20 )
e
0,0500
 r ( 20 50 years )( 20011997 )
e
e
 r (51100 years ) ( 2001 1997 )
Observed
m ortality rates
mortality trend
projected
from 1997 to
2001
0,0300
lower boundary
0,0100
-0,0100 20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98
age
15
Mortality Projections for Life Annuities (example)
As may be seen, the previous graph illustrates several features related to the Portuguese
mortality of male insured lives of the survival-risk-type of life assurance contracts
(basically, endowment, pure endowment and savings type of policies) for the period
between 2000 and 2002:
 The mortality trend for the period 2000-2002 (centred in 2001) is adequately fitted to
the observed mortality data and has been projected from the Gompertz adjusted
mortality trend corresponding to the period between 1995 and 1999, with k=0.05 for
the age band from 20 to 50 years and with k=0.09 for the age band from 51 to 100
years. The parameter r, which translates the annual averaged relative decrease in
the force of mortality for every age assumes two possible values; r=0.05 for the
age band from 20 to 50 years and r=0 for the age band from 51 to 100 years:
 Some minor adjustments to the formulae had to be introduced, for example, the
formula for the force of mortality for the age band from 51 to 100 years is best based
(50 36 )
( Spline1997)
on the force of mortality at age 36, m 36 1
multiplied by a scaling factore
2
than if it were directly based on the force of mortality at age 51:
(Gompertz2001)
m x
1
2
( Spline1997)
= m36 1
2
k (51100 years )( x 36 )
e
 r (51100 years )( 20011997)
e
16
Mortality Projections for Life Annuities (example)
 Further to that, some upper and lower boundaries have also been added to the
graph. Those boundaries have been calculated according to given confidence
levels in respect of the mortality volatility (in this case 1  ( ) = 99,9% and
1  ( ) = 0,1% ) calculated with the normal approximation to the binomial
distribution, with mean E[q x ] = Ex  qx and volatility  [q x ] = Ex  qx  (1  qx )
 The upper boundary may, therefore, be calculated as:
(qx )@99.9% confid. level for q
x
=
(Ex  qx )  [ 
 And the lower boundary may be calculated as:
(qx )@99.9% confid. level for q
x
=
(Ex  qx )  [ 
]
Ex  qx  (1  qx )
subject to 0  qx  1
Ex
]
Ex  qx  (1  qx )
subject to 0  qx  1
Ex
 Those approximations to the normal distribution are quite acceptable, except at
the older ages, where sometimes there are too few lives in E x , the “Exposed-torisk”
17
Mortality Projections for Life Annuities (example)
As for the rest, the process is relatively straightforward:
 From the Exposed-to-Risk ( E x )at each individual age, and from the observed
mortality (q x ) we calculate both
the Central Rate of Mortality ( mx ) and the

Initial Gross Mortality Rate ( q x ) and assess the Adjusted Force of Mortality
)
( m x( Spline
) using “spline graduation”
1

2
(Gompertz )
m
 We then calculate the parameters for the Gompertz( Spline
model
that
produce
x  12
)
in a way that replicates as close as possible the m x  12
 The details of the process are, perhaps, best illustrated in the table presented in
the next page;
 This process has been tested for male, as well as for female lives, so far with
very encouraging results, but we should not forget that we are only comparing
data whose mid-point in time is distant only some 4 or 5 years from each other
and that we need to find a more suitable solution for the upper and lower
boundaries at the very old ages.
18
19
Mortality Projections for Life Annuities (example)
qx
Mortality of insured lives of the survival-risk-type of life assurance in Portugal:
2000-2002 (Males)
0,0100
0,0050
0,0000
20 22 24 26 28 30 32 34 36 38 40 42 44 46 48 50 52 54 56 58 60 62 64 66 68 70
-0,0050
ages
20
Mortality Projections for Life Annuities (example)
As may be seen in the graph below, between the young ages and age 50 there are multiple
decremental causes beyond mortality among the universe of beneficiaries and annuitants
of Pension Funds. That impairs mortality conclusions for the initial rates, which have to
be derived from the mortality of the population of the survival-risk-type of Life Assurance
Mortality of the Beneficiaries and annuitants of Pension Funds in Portugal
2000-2002 (Males)
qx
0,0340
0,0290
0,0240
0,0190
0,0140
0,0090
0,0040
-0,0010
20
22 24 26
28 30
32 34 36
38 40 42
ages
44 46
48 50 52
54 56 58
60 62
64 66 6
21
6. Mortality Projections for Life Annuities (example)
 In general, the mortality rates derived for annuitants have to be based on the
mortality experience of Pension funds’ Beneficiaries and Annuitants from age
50 onwards but, between age 20 and age 49 they must be extrapolated from
the stable trends of relative mortality forces between the Pension Funds
Population and that of the survival-risk-type of Life Assurance.
Annuitants (Males)
0.05( x  t )0.008[t  (t  2006)]
Ages 2040 :
Ages 4149 :
m xt t (T ) = 1.65  0.000144446  e
m xt t (T ) = [1.65  0.0654  (x  t  40)]
0.05( x  t )0.008[t  (t  2006)]
 0.000144446  e
Ages 50   :
m
t
x t
0.09( x  t )0.0075[t  (t  2006)]
(T ) = 0.000044994  e
Where T is the Year of Projection and 2006 is the Reference Base Year
22
6. Mortality Projections for Life Annuities (example)
Annuitants (Females)
0.028( x  t ) 0.008[t  (t  2006)]
Ages 2034 :
Ages 3544 :
m xt t (T ) = 1.415  0.000211957  e
m xt t (T ) = [1.415  0.10495  (x  t  35)]
0.028( x t )0.008[t  (t  2006)]
Ages 45   :
m
 0.000211975  e
t
x t
0.09( x  t )0.0075[t  (t  2006)]
(T ) = 0.000033095  e
Where T is the Year of Projection and 2006 is the Reference Base Year
 The above formulae roughly imply (for both males and females) a Mortality
Gain (in life expectancy) of 1 year in each 10 or 12 years of elapsed time, for
every age (from age 50 onwards).
23
Mortality Projections for Life Annuities (example)
Annuitants
 As was mentioned before, for assessing the mortality rates at the desired
confidence level we may use the following formulae:
qx 
m x
1
2
1  12  m x  1
 Ex  qx    Ex  qx  (1  qx ) 
(qx )@ confidence level α = max 
,
Ex


where α is such that Φ(α ) = desired confidence level
2

0

In our case ()=99,5% which implies that   2,575835
 Now, to use the above formulae we need to know two things:  The dynamic
mortality trend for every age at onset, and  the numeric population
structure.
24
Mortality Projections for Life Annuities (example)
 In order to
calculate the trend
for the dynamic
mortality
experience of
annuitants we
need to use the
earlier mentioned
formulae and
construct a
Mortality Matrix:
25
Mortality Projections for Life Annuities (example)
 In order to calculate a Stable Population Structure we need to smoothen the
averaged proportionate structures from several years experience
Population Structure of Pension Funds Beneficiaries and
Annuitants (Males: Exposed-to-Risk) Projection for 2006
8.000
3-year Exposed-to-Risk
7.000
6.000
20 year Moving
Average for the 3year Exposed-toRisk
5.000
4.000
3.000
1-year Exposedto-Risk (stable
structure)
2.000
1.000
99
90
81
72
108
ages
63
54
45
36
27
18
9
0
0
26
Mortality Projections for Life Annuities (example)
 We are now able to project the dynamic mortality experience for different
ages at onset and for different confidence levels
27
Supervisory process
ISP analysis of suitability of mortality tables used
• ISP receives annually information regarding the mortality
tables used in the calculation of the mathematical provisions
• This information is compared with the overall mortality
experience of the market and with mortality projections
• ISP makes recommendations to actuaries and insurance
companies to reassess the calculation of mathematical
provisions with more recent tables whenever necessary
28
Mortality Projections for Life Annuities (example)
rtz 2 004 )
( Spline 1998 )
m x(Gompe
= m37
 e ( 59 100 anoss )
1
1
k
2
2
 (x  57 )
e
r (59100 anos ) ( 2004 1998 )
qx
rtz 2004 )
( Spline 1998 )
m x(Gompe
= m37
e


1
2
1
2
k ( 30 58 anos )  ( x  37 )
e
 r( 30 58 a nos )  (2004 1998 )
0,3400
0,2900
0,2400
2004
0,1900
0,1400
2001
0,0900
0,0400
1998
-0,0100
20 23 26 29 32 35 38 41 44 47 50 53 56 59 62 65 68 71 74 77 80 83 86 89 92 95 98
Idades
(x)
Ages (x)
29
Statistical Quality Tests for Mortality Projections
Idade
Actuarial
YEARS: 1997/1999
PENSION FUND PENSIONERS (MALES)
Males
(2)
(3)
(4)
"Exposed to Risk"
Observed
Projected
Ex
Mortality
Mortality
(X)
3
Splines
13,9
17,1
19,9
23,6
29,2
35,3
41,8
46,1
51,2
58,3
63,3
68,7
72,2
78,5
83,1
105,7
115,5
123,2
129,6
131,5
139,6
144,9
150,5
158,2
160,8
166,5
169,1
168,9
154,4
136,3
133,9
140,5
140,1
130,6
119,7
114,8
106,4
98,1
89,7
77,1
63,5
4.071,1
n=
38
Number of
Number of Variables
Number of Degrees of
elements in
parametrised in the
in the Chi-squared
the age band
adjustement
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Sum
41
m x(Gompertz

1
2
1.630,0
2.062,5
2.411,0
2.820,5
3.355,0
3.985,0
4.681,0
5.010,0
5.278,0
5.595,0
5.809,5
6.115,0
6.015,0
5.926,0
5.552,5
6.369,0
6.413,0
6.244,5
5.951,8
5.434,0
5.226,5
4.956,5
4.693,5
4.485,5
4.137,0
3.844,5
3.512,5
3.190,5
2.678,5
2.190,0
1.955,0
1.837,5
1.665,0
1.428,5
1.220,5
1.063,0
872,0
721,5
601,0
481,5
364,5
1998 )
= m 37( Spline
1
2
1998 )
8,0
16,0
24,0
23,0
33,0
27,0
34,0
35,0
65,0
41,0
70,0
67,0
69,0
87,0
83,0
121,0
131,0
127,0
124,0
115,0
150,0
138,0
154,0
137,0
139,0
157,0
184,0
168,0
155,0
160,0
136,0
152,0
125,0
141,0
147,0
136,0
117,0
116,0
113,0
100,0
74,0
4.199,0
e
Distribution
k ( 59  100 anos ) ( x  57 )
(5)
Projected
Mortality
Gompertz
15,0
19,2
22,6
26,7
32,1
38,5
45,7
49,4
51,5
56,4
64,0
73,7
79,2
85,4
87,5
109,7
120,7
128,5
133,9
133,6
140,5
145,6
150,6
157,3
158,4
160,8
160,5
159,1
145,8
130,1
126,8
130,0
128,4
120,1
111,8
106,1
94,8
85,4
77,4
67,4
55,5
3.985,9
Actuarial
Age
(X)
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
Squared
Deviations
of Mortality
Desviations
sum[(3)-(4)]
Contribution to
the Chi-squared
Test
Splines
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Sum
-
-
-
0
+
+
-
-
+
+
-
-
-
0
-
0
+
+
-
-
+
+
-
-
-
0
+
+
-
-
+
+
+
0
+
0
-
-
-
0
+
+
-
-
+
+
-
-
-
0
-
0
+
+
-
-
+
+
+
0
+
0
+
0
-
-
+
+
+
0
+
0
+
0
+
0
+
0
+
0
+
 =
Value of the Chi-sqr  n2 =
0
23
18
Groups
of signs
(+)
(-)
n=
tk =
Value of the Chi-sqr =
n=
Distribution between
age 50 and 80 at t k =
the Quantile t k
2
n
11
11
Prob.gr.desv.(+)>g(+)=
Idade
Actuarial
72,82812
Splines
(q x ) 
 corr
38
0,0574%
40,40533
28
6,0824%
Gom pertz
(q x )
 corr
=
Splines
(q x )
 corr
1,332777438 E  q x
1,338541933 
1
(99 ,9% )
1,338542   1 (99 % )  1,338542  2,32634  3,11390
Squared
Deviations
of Mortality
Desviations
sum[(3)-(5)]
(X)
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
####
2
n
Distribution at
Quantile t k
Signs
(+)
(-)
Males
Males
Confidence
Confidence
Approximate
Level of
Level of
Std. Deviation
Variance
Mortality at
Mortality at
[(3)-(4)]/sqr(4)
sum[(4)]
( = 2,3263 ) ( = 2,3263 )
99,0%
1,0%
Splines
Splines
Gompertz
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
Sum
-
-
-
0
+
+
-
-
+
+
-
-
-
0
-
0
+
+
-
-
+
+
-
-
-
0
+
+
-
-
+
+
+
0
-
-
-
0
-
0
+
+
-
-
+
+
-
-
-
0
-
0
+
+
+
0
+
0
+
0
+
0
+
0
-
-
+
+
+
0
+
0
+
0
+
0
+
0
+
0
+
0
2
Signs
(+)
(-)
Valor Distribuição  =
n=
do Qui-quadrado
entre 50 e 80 anos t k =
no Quantil t k
2
n
Groups
of signs
10
10
41,4287%
(-oo, -3)
0
0
Ranges
Observ.Dev.
(
(-3, -2)
0,82
1

q  Ex  qx b
 x
Ex  qx b
x
min 
2n


)
2

 n
=



(-2, -1)
5,74
8
Approximate
Std. Deviation
[(3)-(4)]/sqr(4)
Approximate
Variance
sum[(4)]
Gompertz
Gompertz
Prob.gr.desv.(+)>g(+)=
130,48575
38
Gompertz
(q x ) 
 corr
0,0000%
1,783978488 E  q x
49,16566
28
0,8001%
66,1878%
(Stevens Test)
Standard Deviations'
Deviations' Test
Test
Standard
(-1, 0 )
( 0, 1 )
13,94
13,94
9
11
( 1, 2 )
5,74
8
( 2, 3 )
0,82
4
( 3, oo)
0
0
(-oo, -3)
0
0
Ranges
Observ.Dev.
Observ.Dev.
(
3,26383
Males
Males
Confidence
Confidence
Level of
Level of
Mortality at
( = 3,09 ) Mortality
( =  3,at
09 )
99,9%
0,1%
 n2 =
Valor Distribuição  n =
do Qui-quadrado no n =
tk =
Quantil t k
(Stevens Test)
Observ.Dev.
Contribution to
the Chi-squared
Test
 b=
1,03960

q  Ex  qx b
 x
Ex  qx b
x
min 
2n


(-3, -2)
0,82
2
)
2

 n
=



(-2, -1)
5,74
7
8,52283
Standard
Standard Deviations'
Deviations' Test
Test
(-1, 0 )
( 0, 1 )
13,94
13,94
9
10
 b=
( 1, 2 )
5,74
5
( 2, 3 )
0,82
4
( 3, oo)
0
4
1,06750
30
Mortality Projections: Variance Error Correction
2004
YEARS: 2003/2005
PENSION FUND PENSIONERS (MALES)
(1)
Age
(2)
"Exposed to Risk"
x
(3)
Observed
Mortality
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
Sum
38
(4)
Expected
Mortality
Gompertz
7.918,5
8.284,0
8.353,0
8.150,5
7.762,0
7.464,0
7.306,5
8.269,0
9.081,0
8.625,0
8.039,5
7.353,0
7.825,0
8.747,0
8.351,5
7.772,5
6.883,0
5.918,5
5.144,5
4.524,5
4.127,0
3.674,5
3.198,0
2.829,0
2.490,5
2.129,5
1.769,5
1.463,5
1.242,0
1.044,0
838,0
670,5
537,5
413,5
310,5
242,5
n
171,0
122,0 q x =
x
5
98,0
118,0
142,0
176,0
149,0
125,0
170,0
201,0
195,0
179,0
181,0
180,0
204,0
226,0
279,0
238,0
248,0
207,0
223,0
198,0
226,0
196,0
218,0
215,0
206,0
196,0
143,0
157,0
164,0
144,0
149,0
111,0
93,0
86,0
72,0
60,0
= 46,0
38,0
6.257,0
33
Number of
Number of Variables
Number of Degrees of
elements in
parametrised in the
in the Chi-squared
the age band
m x(Gz om pert
adjustement
2 004 )
1
2
(Spline 1998 ) 
= m 37
e
1


  n  2   E xt  q xt  
x


(
bt
k (59  100 a n os s )( x  57 )
2
)
x
73,8
79,8
88,0
93,9
97,8
102,8
110,1
136,2
163,5
169,8
173,0
173,0
201,2
245,8
256,4
260,7
252,2
236,9
224,9
216,0
215,1
209,0
198,5
191,6
183,9
171,5
155,3
140,0
129,4
118,4
103,5
90,1
78,5
65,6
53,5
45,4
34,7
26,8 =
5.566,5
(
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
t
E x95
 q xt
1
 n
x
(q
t
x
 E xt  q xt )
E xt  q xt
2
)
Cummulative Deviations
in Mortality
-r
 (q
(59 100 a n os ) (2004  1998 )
)

  (E xt  q xt ) 

(
(E
9,9365985 E x  q x
t
(q x  E x  q x )2
 Ex  qx )
Ex  qx
583,7
7,9
1.460,7
18,3
2.918,0
33,2
6.742,7
71,8
2.623,7
26,8
492,0
4,8
3.593,3
32,7
4.200,8
30,8
991,1
6,1
84,8
0,5
63,9
0,4
49,6
0,3
8,0
0,0
390,4
1,6
510,6
2,0
516,3
2,0
17,9
0,1
894,2
3,8
3,6
0,0
322,9
1,5
119,6
0,6
169,1
0,8
380,3
1,9
548,9
2,9
486,4
2,6
600,4
3,5
152,2
1,0
289,2
2,1
1.196,3
9,2
653,1
5,5
2.072,6
20,0
438,3
4,9
210,2
2,7
415,0
6,3
341,5
6,4
214,1
4,7
127,7
3,7
124,7
4,6
35.007,8
327,9
Value of the Valor
Chi-sqr
da Distribuição
Value of
of the
the Chi-sqr
Chi-sqr
Value
Distribution
doat
Qui-quadrado Distribution
no
Distribution at
at
Quantile t k 2 Quantil t k Quantile
(q  E tkq )2
x (q x  E x  q x ) F 2 (t k ) = x x E xq x
x
x
(8)
Readjusted
Expected
Mortality
Ex  qx  bt
84,4
91,2
100,6
107,3
111,8
117,5
125,8
155,7
186,9
194,1
197,8
197,7
230,0
281,0
293,1
298,1
288,4
270,8
257,1
246,9
245,9
238,9
226,9
219,0
210,3
196,1
177,6
160,0
147,9
135,4
118,3
103,0
89,7
75,0
61,2
51,9
39,7
30,7
6.363,8
t
t
x
(
q t  E t  qt
 
 q xt     E xt  q xt   x t x t x
Ex  qx
x
  x
)
(
)
)
)
2
b1 =
b =
2
1,03472
0,95672
q x  Ex  qx bt
13,6
26,8
41,4
68,7
37,2
7,5
44,2
45,3
8,1
-15,1
-16,8
-17,7
-26,0
-55,0
-14,1
-60,1
-40,4
-63,8
-34,1
-48,9
-19,9
-42,9
-8,9
-4,0
-4,3
-0,1
-34,6
-3,0
16,1
8,6
30,7
8,0
3,3
11,0
10,8
8,1
6,3
7,3
-106,8
(q
x
 Ex  qx  bt
)
2
(7)
Contribution to
the Chi-squared
Test
2
q x  Ex  qx  bt
Ex  qx  bt
(
)
184,5
717,8
1.715,3
4.714,9
1.385,0
55,6
1.951,9
2.052,6
65,0
228,3
281,8
314,5
675,5
3.020,6
199,7
3.608,3
1.628,9
4.075,1
1.163,0
2.391,6
394,8
1.843,9
79,8
16,1
18,4
0,0
1.196,3
9,3
257,6
73,8
942,8
64,6
10,6
120,4
116,9
66,1
40,1
53,7
35.734,9
2,2
7,9
17,1
43,9
12,4
0,5
15,5
13,2
0,3
1,2
1,4
1,6
2,9
10,8
0,7
12,1
5,6
15,0
4,5
9,7
1,6
7,7
0,4
0,1
0,1
0,0
6,7
0,1
1,7
0,5
8,0
0,6
0,1
1,6
1,9
1,3
1,0
1,7
213,7
Cummulative Deviations
Value of the Chi-sqr
in Mortality
Distribution at
Quantile
(q  Etxk q x )2
2
F 2 (t k ) =  x
x q x  E x  q x  b t
Ex  qx
x
)
tk =
0,0000%
 

 
(10)=(9)2
Squared
Desviations in
Mortality
(9)=(3)-(8)
Desviations in
Mortality
(
tk =
2
 

t
t 
 n  2   E x  q x   4  
x

 x
 

t
2    E x  q xt
 x
(
x
x
(
2
(q xt )    
 corr
24,2
38,2
54,0
82,1
51,2
22,2
59,9
64,8
31,5
9,2
8,0
7,0
2,8
-19,8
22,6
-22,7
-4,2
-29,9
-1,9
-18,0
10,9
-13,0
19,5
23,4
22,1
24,5
-12,3
17,0
34,6
25,6
45,5
20,9
14,5
20,4
18,5
14,6
11,3
11,2
690,5
x
(7)
Contribution to
the Chi-squared
Test
(q x  E x  q x )2
q x  Ex  qx
Distribution
e
(6)=(5)2
Squared
Desviations in
Mortality
(5)=(3)-(4)
Desviations in
Mortality
Idade
Ex  qx
qx
Ex
T=2005-1998= 6
TIME HORIZON OF MORTALITY PROJECTION
0,0000%
b16 =
b26 = 1,22723
0,76685

)
 corr (q xt )  9,9365985  E xt  q xt  3,1522371  E xt  q xt
q xt (1   (a )) =  q1 (  ) = (E xt  q xt )     corr (q xt )  (E xt  q xt )  (3,1522371   )  E xt  q xt
t
x
q xt ( ( )) =  q 1 ( ) = (E xt  q xt )     corr (q xt )  (E xt  q xt )  (3,1522371   )  E xt  q xt
t
x
31
Pension Fund Beneficiaries (Males 2003-2005)
Nº
400
confidence leval at 1% (readjusted)
350
300
Observed values
250
200
Fitted values
150
100
confidence level at 99%
(readjusted)
50
Non-adjusted
Confidence Levels
94
92
90
88
86
84
82
80
78
76
74
72
70
68
66
64
62
60
58
0
age
32
33